**Exploration**

The dynamic figure below follows up on the Matric Exam Geometry Problem - 1949, and will help you find how to determine the maximum of ∠*BEC*. It shows a circle constructed with its centre *I* on the perpendicular bisector of *BC*, and tangent to the line segment *AD* at *G*.

1) Drag *I* till the circle passes through *B* and *C*.

2) Next drag *E* to coincide with *G* while carefully watching what happens to ∠*BEC*. What do you notice?

3) Change the shape of the trapezium *ABCD*, repeat the preceding process, and check your observation again.

**Challenge**: Can you explain (prove) why the maximum of the angle has to be located at the position of *E* you identified?

4) Can you figure out a way via construction, not using experimentation and dragging as with this sketch, to precisely locate where the centre *I* of the circle should be located to determine a maximum for ∠*BEC*?

Maximizing the Angle

*Hints*

5) Try using the exterior angle theorem to prove why the maximum of angle *BEC* is located where it is.

6) Consider the locus of points equidistant from vertex *B* and line *AD* and/or the locus of points equidistant from vertex *C* and line *AD*.

7) Carefully reflect on your solution/proof. Is it necessary that *AB* // *CD*? Can you generalize further?

8) Click on **Show Solution** button to display a constructed solution.

**Published Paper**

Check your explorations above, by reading my 2015 paper in *At Right Angles* and *Learning and Teaching Mathematics* at Flashback to the Past: a 1949 Matric Geometry Question.

**Earlier References**

I'm grateful to Vladimir Dubrovsky, Moscow, Russia, who recently (Jan 2024) on *Facebook* pointed out that the maximum angle problem above appeared earlier in the so-called "Bus Problem" of Chapter 5 of the following publication: Vasilyev, N.B. & Gutenmacher, V.L. (1980). Straight Lines & Curves. Mir Publishers, Moscow.

The problem also directly relates to maximizing the place kicking angle in rugby (De Villiers, 1999).

**Related Papers**

De Villiers, M. (1998). Exploring loci on Sketchpad. *Pythagoras*, 47, Dec., pp. 71-73.

De Villiers, M. (1999). Place Kicking Locus in Rugby. *Pythagoras*, 49, Aug., pp. 64-67.

**Some Related Links**

Matric Exam Geometry Problem - 1949

All parabola are similar - i.e. have the same shape

Cyclic Hexagon Alternate Angles Sum Theorem

A generalization of the Cyclic Quadrilateral Angle Sum theorem

Angle Divider Theorem for a Cyclic Quadrilateral

Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi

Alternate sides cyclic-2n-gons and Alternate angles circum-2n-gons

Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral

An extension of the IMO 2014 Problem 4

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Created by Michael de Villiers, 14 August 2015 with *WebSketchpad*; updated 19/27/28 Jan 2024.